7,526 research outputs found
Smoothed analysis of the low-rank approach for smooth semidefinite programs
We consider semidefinite programs (SDPs) of size n with equality constraints.
In order to overcome scalability issues, Burer and Monteiro proposed a
factorized approach based on optimizing over a matrix Y of size by such
that is the SDP variable. The advantages of such formulation are
twofold: the dimension of the optimization variable is reduced and positive
semidefiniteness is naturally enforced. However, the problem in Y is
non-convex. In prior work, it has been shown that, when the constraints on the
factorized variable regularly define a smooth manifold, provided k is large
enough, for almost all cost matrices, all second-order stationary points
(SOSPs) are optimal. Importantly, in practice, one can only compute points
which approximately satisfy necessary optimality conditions, leading to the
question: are such points also approximately optimal? To this end, and under
similar assumptions, we use smoothed analysis to show that approximate SOSPs
for a randomly perturbed objective function are approximate global optima, with
k scaling like the square root of the number of constraints (up to log
factors). Moreover, we bound the optimality gap at the approximate solution of
the perturbed problem with respect to the original problem. We particularize
our results to an SDP relaxation of phase retrieval
Sparse Recovery of Positive Signals with Minimal Expansion
We investigate the sparse recovery problem of reconstructing a
high-dimensional non-negative sparse vector from lower dimensional linear
measurements. While much work has focused on dense measurement matrices, sparse
measurement schemes are crucial in applications, such as DNA microarrays and
sensor networks, where dense measurements are not practically feasible. One
possible construction uses the adjacency matrices of expander graphs, which
often leads to recovery algorithms much more efficient than
minimization. However, to date, constructions based on expanders have required
very high expansion coefficients which can potentially make the construction of
such graphs difficult and the size of the recoverable sets small.
In this paper, we construct sparse measurement matrices for the recovery of
non-negative vectors, using perturbations of the adjacency matrix of an
expander graph with much smaller expansion coefficient. We present a necessary
and sufficient condition for optimization to successfully recover the
unknown vector and obtain expressions for the recovery threshold. For certain
classes of measurement matrices, this necessary and sufficient condition is
further equivalent to the existence of a "unique" vector in the constraint set,
which opens the door to alternative algorithms to minimization. We
further show that the minimal expansion we use is necessary for any graph for
which sparse recovery is possible and that therefore our construction is tight.
We finally present a novel recovery algorithm that exploits expansion and is
much faster than optimization. Finally, we demonstrate through
theoretical bounds, as well as simulation, that our method is robust to noise
and approximate sparsity.Comment: 25 pages, submitted for publicatio
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